Solutions to Electrodynamics (2)

Solutions (2):

1) We have:   B = E /c

   =   (220 V/ m)  /  (3 x 10 ⁸ m/s) =   7.3  x 10^-7  T

2)      The intensity of an EM wave is :


I = S(av) = E(max) B(max)/ 2 m_o


where the area A = 2.4 m x 0.7 m = 1.68 m ²  and P _R  denotes the radiation pressure, or   P _R = S/c (for complete absorption)


Now, S = Power/ area )=  ( 25 W/m ² ) / (1.68 m ²) = 14.88 W


And the radiation pressure:


P _R = S(av) / c = 14.88 W/ (3 x 10 ⁸ m/s)


= 3.3 x 10  ^{- 9}  N/m²



(3) We have: P(av) = 4 kw = 4 x 10 ³ W


Assume spherical symmetry for area affected so area of the applicable spherical surface is:  4 π r ²

With r the distance (radius) to  the radio station transmitter but convert to m :

r  = 4 mi. = 

 4 mi (5280 ft./mi)(0.303 m./ft)  =    6.4 x 10³   m 


Area = 4 π ( 6.4 x 10³  m) ² = 5.1  x 10 ⁸ m²


Therefore the average Poynting vector associated with the transmission is:


S(av) = P(av)/ A =

 (4 x 10 ³ w)/ (5.1  x 10 ⁸ m²   ) =  7.7 x 10 ^-6  W/m²


But recall:


S(av) =  S(av) = E(max) ²/ 2 m_o c

Therefore, solving for E(max):


E(max) = [2S(av) m_o c] ^½


Then: E(max)=

 [2(7.7 x 10 ^-6  W/m² ) (4 π x 10 ^-7  H/m)(3 x 10 ⁸ m/s)] ^½


E(max) = 0.076   V/m


Then the emf induced in a 65 cm long (L =0.65m) antenna is:


Emf = E(max) L = (0.076 V/m) x (0.65m) = 0.049  V



(4) We assume: P(solar) = 10 ³ W/m²


But because efficiency is relevant we need P(in). Thus,


eff = P (out)/ P(in) = 0.3 = 1 MW/ P(in)


Where 1 MW = 10 ⁶ watts is the desired energy to come out, or be produced.


To get this, the power we need to put in, is:


P(in) = P(out)/ eff = (10 ⁶ w)/ 0.3 = 3.33 x 10 ⁶ W


Then:

Area A = P(in)/ S =

 (3.33 x 10 ⁶ W)/ (10 ³ W/m²)    = 3.33 x  10 ³  m²

Or about 3, 330 square meters.

5 ) By definition of current: I = nevA


where v is the drift velocity: v = I/neA

Where A = z1 y1 = (2 cm) (1 mm) = 0.02m (0.001 m) =

(2 x 10  ^-2 m) (1 x 10  ^-3 m) = 2 x 10  ^-5 m ²

Therefore:

v = (200A)/ [(7.4 x 10 ²⁸ /m )(1.6 x 10 ^-19 C)(2 x 10  ^-5 m ²)]

v = 8.4 x 10 ^-4 (-x^) m/s


b) E = V_H / z1

where z1= 0.02m (= 2 cm)

But: V_H = BI/ new

= (1.5T) (200 A) / [(7.4 x 10 ²⁸ /m )(1.6 x 10 ^-19 C)( 1 x 10  ^-3 m)]

= 2.5 x 10  ^-5   V (-z^)

Then: E =  V_H / z1  = 

2.5 x 10  ^-5  V (-z^)/ 0.02 m = 1.2 x 10  ^-3  V/m(-z^)


c)  N.B. The answer is already found above as V_H = BI/ new